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TECHNICAL PAPERS: Soft Tissue

A Constituent-Based Model for the Nonlinear Viscoelastic Behavior of Ligaments

[+] Author and Article Information
P. Vena

Department of Structural Engineering,  Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Milan, Italyvena@stru.polimi.it

D. Gastaldi, R. Contro

Department of Structural Engineering,  Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Milan, Italy

J Biomech Eng 128(3), 449-457 (Dec 12, 2005) (9 pages) doi:10.1115/1.2187046 History: Received July 26, 2005; Revised December 12, 2005

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista, 2005, J. Orthopaed. Res., 23, pp. 67–76;Hingorani, 2004, Ann. Biomed. Eng., 32, pp. 306–312;Provenzano, 2001, Ann. Biomed. Eng., 29, pp. 908–214;Weiss, 2002, J. Biomech., 35, pp. 943–950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Strain dependence of the relaxation ratio: the dashed line refers to experimental results on human ligaments; the solid line refers to experimental results on rabbit ligaments

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Figure 2

Cauchy longitudinal stress for a uniaxial tension along the direction parallel to the fibers; experimental results on rabbit MCL (14) and curve fitting with the proposed model

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Figure 3

Coefficient α as a function of the strain for the identified model of rabbit MCL

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Figure 4

Stress relaxation factor at 100s; dots represent experimental results (14), the solid line represents the model prediction

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Figure 5

Relaxation rate nR from 36 as a function of the applied longitudinal strain; dots represents the experimental results on rabbit MCL (14), the solid line represents the model prediction

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Figure 6

Creep rate nR from 37 as a function of the applied stress; dots represent the experimental results (14), the solid line represents the model prediction

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Figure 7

Cauchy stress in longitudinal and transverse directions in human MCL; symbols represent the experimental results (17), the solid line represents the curve fitting with the proposed model

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Figure 8

Coefficient α as a function of the strain for the identified model of human MCL

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Figure 9

Relaxation rate as a function of the applied longitudinal strain; dots represent the experimental results on human MCL (17), the solid line represents the model prediction

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